User blog:Vel!/IRC log 2014-11-23
A pretty good (if heated) conversation today about BEAF and formality. 10:34:03 previously you discussed several attempts at extensions in xE^ 10:34:09 Do most ppl still think legion arrays are LVO? 10:34:12 aarex's, iko's, vell's, holloms (?) 10:34:18 Bowers' still seems to be under that impression 10:34:23 hm that's odd 10:34:29 they're usually believed to be psi(W_w) 10:34:35 i know 10:34:49 but remember, Bowers' didn't start out knowing anything about ordinals 10:34:57 Who believes that they reach psi(W_w), actually? 10:35:05 and is only know incorporating that into his work on BEAF 10:35:06 it's the most common belief 10:35:14 hyp cos used it in his analysis 10:35:22 the wiki had that on the fgh's page 10:36:23 unlike with xE^, whatever the linear-array-e is named will be quite hard to logically extend 10:37:04 SbiisExE: in your xE^ article you say that hollom talked about an extension 10:37:10 but the link couldn't take me to it 10:37:55 you talking about the one where (#^^a)^^b = #^^(a+b) 10:38:15 @Wojo Who believe it reaches psi(W_w) actually? Me for one 10:38:41 it doesnt reach anything. its undefined 10:38:47 Thank you Sbiis for more precise answer than cf's 10:38:58 I believe the LVO result is an artifact of taking the ordinal-arithmetic too literally 10:39:20 so there is no catching point for SGH and FGH? 10:39:23 --> Flitris (xxxxxxx@gateway/web/freenode/ip.address censored) has joined ##googology 10:39:24 Hello Flitris! 10:39:29 Hey Flit 10:39:31 professional mathematicians are talking about something ill-defined? 10:39:39 Sbiis, depends on fundamental sequences 10:39:43 of course vell 10:39:43 right 10:39:45 so what 10:39:46 oh my lotsa people here today 10:39:53 yeah 10:39:58 for a chosen set of fundamental sequences, the question becomes meaningful 10:39:59 then 10:40:03 Exactly 10:40:14 And it has been answered for few such choices 10:40:22 which means there is nothing inherently impossible about defining BEAF 10:40:29 Of course not 10:40:30 It is possible in principle 10:40:34 Of course it is 10:40:43 Wojowu: remember your fgh vs. peano arithmetic question? did you ever get an answer for the f_a > f_b restriction? 10:40:58 For example: I define all arrays beyons pentational to evaluate to 1. Here, I gave you a definition 10:41:10 No, I didn't, vell 10:41:40 ok that's just smartass 10:41:48 So? 10:41:48 cookiefonster: reality is smartass 10:41:52 It's a definition 10:42:02 it doesn't help the discussion, it's nothing more than smartass 10:42:02 Maybe a bad one, but this is subjective now 10:42:14 It's still a valid defintiion 10:42:16 cookiefonster: it does help the discussion, because it proivdes a counterexample to a claim 10:42:31 but it's smartass 10:42:32 as I've said before 10:42:36 Counterexample to what claim? 10:42:44 something within Bowers' parameters 10:42:47 not just ANY definition 10:42:56 why in the heck would I argue for something so trivial 10:42:57 the implicit claim that any definition of BEAF beyond e0 makes intuitive sense 10:43:04 you know that's not what I mean 10:43:17 SbiisExE: okay then what are bowers parameters 10:43:25 when we treat everything as formal we get this smartass shit 10:43:31 which is why we can't treat everything as foral 10:43:35 formal * 10:43:45 9_9 10:44:26 obviously an array should not evaluate to 1 for all values 10:44:33 Well, I gave one definition, you can think of it as a shit definition 10:44:40 unless the prime and base equal 1 of coarse 10:44:44 But I didn't say we can't have better definitions 10:45:01 or just the base = 1 10:45:03 SbiisExE: okay then what are bowers parameters 10:45:10 Every new structure should contain every previous structure as a substructure 10:45:25 define a structure 10:45:29 so if we have an array of tetrational spaces then 10:45:45 that's the kind of thing that goes without saying 10:45:53 cookiefonster: it sure as hell does not 10:45:55 {3,3 X 2 } = { 3^^3 & 3 X 3^^3 & 3 X 3^^3 & 3 } 10:45:56 unless you're a smartass of course 10:46:04 X ? 10:46:07 without a precise definition of "structure" there is room for loopholes and smartassery 10:46:08 this would obviously have exactly 3^^3 * 3 non-1 entries 10:46:17 the point of formality is to EVADE this smartassery you speak of 10:46:22 Okay, then at least specify a way of determining if a structure is a substructure of another one 10:46:27 It should also be larger than { 3^^3 & 3 } 10:46:38 so how could it be 1 under any REASONABLE definition? 10:46:54 "define reasonable definition 10:47:03 <-- Flitris (xxxxxxxx@gateway/web/freenode/ip.address censored) has quit (Quit: Page closed) 10:47:06 Did I ever say my definition is reasonable in any way? 10:47:38 what wojowu gave is, indeed, a definition 10:47:50 and since we have no idea what "bowers' parameters" are 10:47:58 (sicne he never formally defined shit) 10:47:59 A structure is an ordered set of entries obviously 10:48:06 an ordered set of entries? okay 10:48:33 where "entry" is defined as a position in the array 10:48:37 we can determine the structure being used by finding the pilot 10:48:48 well, we can at least give parameters with ... common sense 10:48:50 {b,p(1)2} 2 is a pilot at position "w" 10:49:04 So you assume positions are indexed with ordinals? 10:49:05 this expands the "w" structure, called a linear array by Bowers' 10:49:17 i define my structure as the following ordered set of entries: the seventh, the sixth, the eleventh, and then the nineteenth 10:49:21 then we define it's growth rate by how many entries are produced when prime = p 10:49:34 {b,p (1) 2 } = {b,b,...,b} w/p b's 10:49:45 therefore w-structure has growth rate p 10:49:56 w^2-structure has growth rate p^2 10:49:58 and so on 10:50:17 let a comma be the prime delimiter 10:50:30 a successor delimiter forms an array of the previous structure 10:50:38 and then we have limit delimiters/structures 10:50:53 okay lets look at what we DO agree on about BEAF, namely what it is up to e0 10:50:56 such as dimensional-structures or tetrational-structures 10:51:03 not that any of this is new to anyone 10:51:32 can we agre on this definition: http://googology.wikia.com/wiki/Introduction_to_BEAF#As_ordinals_.28advanced.29 10:52:14 to the best of my knowledge it is perfectly formal, although nobody has yet proven that it terminates 10:52:37 (shouldnt be hard though, considering how simple wojowu's proof for linear arrays is) 10:54:11 @Vel What Wojo gave was indeed a definition. Like I said, no one is going to argue against that. 10:54:24 but no one is trying to defend that strawman claim 10:54:33 When I say BEAF can be defined in principle 10:54:44 I mean in the spirit of BEAF as presented by Bowers 10:54:51 i dont feel no spirit 10:54:56 not just ANY definition. That's trivial and is of no interest to anyone 10:54:59 can we agre on this definition: http://googology.wikia.com/wiki/Introduction_to_BEAF#As_ordinals_.28advanced.29 10:55:09 does everyone agree w/ this? 10:55:09 exactly 10:55:10 except those who wish to derail the investigation 10:55:42 i'm trying to get us to reach an agreement on the basics of how BEAF is defined 10:55:58 can everyone say whether they agree or disagree w/ the given definition up to e0 please? 10:56:06 sure wh not 10:56:30 (just fixed a minor error) 10:57:20 @Vel fundamentally no. It uses an approach which is not inline with the climbing method, and has, once again, more to do with ordinal arithmetic 10:57:37 what the fuck is the climbing method?! 10:57:46 glad you asked 10:57:56 who came up with this 10:57:59 It's something that Bowers' mentions on his hypernomials page 10:58:10 "mentions" 10:58:11 and references frequently when discussing going beyond tetrational arrays 10:58:12 and it's what sbiis uses in exe 10:58:20 right 10:58:24 I've said it a few times too 10:58:41 but somehow all of this isn't even a blip on the map even though ... we are all talking about BEAF, no? 11:00:18 BONG BONG BONG BONG BONG BONG BONG BONG BONG BONG BONG 11:00:34 this "climbing method" is meaningless to me http://polytope.net/hedrondude/hypernomial.htm 11:00:48 right off the bat I see X^X^...^X^X "w/ X X's" 11:00:54 It's not really all that meaningless 11:01:07 bowers has discussed that in emails with sbiis 11:01:08 it's true that if you took a power tower of the form p^^p 11:01:14 and you kept multiplying by p 11:01:24 He doesn't even give a description of climbing method 11:01:28 eventually you could observbe the "climb" as Bowers' suggests 11:01:41 we have no idea what "X" means, let alone what "X^X" or "X^^X" means 11:01:46 He just gives an example and says "this I'll call climbing methid" 11:02:09 and i dont see how my ordinal definition " is not inline with the climbing method" 11:02:36 all i see in bowers' discussion of the climbing method is a shit-ton of ellipses and meaningless symbols 11:03:30 sbiis gave a more extensive interpretation of this climbing method as we know 11:03:33 using a > instead of a , 11:04:33 It's not in line for a very simple reason 11:04:43 what is w^^(w2) in your system 11:04:58 how many entries would {b,p ( w^^(w2) ) 2 } actually produce? 11:05:03 my system only goes up to e0 11:05:05 vell asked only if we agree with this up to e0, didn;t he? 11:05:10 exactly p^^(p*2) entries or not? 11:05:19 true 11:05:24 now that kind of thing is where it gets troublesome 11:05:24 SbiisExE: that's actually impossible because Bowers is wrong about the entry count 11:05:31 but I thought his definition was made to go up to legion space? 11:05:37 there's a multitude of interpretations 11:05:41 no 11:05:52 because as you guys love to point out it depends 11:06:02 on the fundamental sequences 11:06:11 the old "prime block of an X structure has as many entries as you get when replacing X's with p's" thing 11:06:33 that kind of thing ... just doesn't work 11:06:36 your fundamental sequences do not produce the required entries 11:06:36 it depends on intuition gained from the Wainer hierarchy, and it just doesnt work 11:06:40 and after the svo, first law of googology 11:06:43 and so obviously Bowers' would never use them 11:07:24 wrong 11:07:31 It can be done 11:07:50 It just depends how you stack things up 11:08:07 you could in theory get it to stack up to be something like X-->X-->X-->X 11:08:10 okay well 11:08:22 but Bowers' idea is to stack it up in a way to get BEAF like numbers 11:08:37 hear me out here: i linked to my definition because i wanted us to agree on basic notions of what structures, entries, arrays, etc. are 11:08:53 but obviously that's not even going to wrok 11:09:39 bowers has doen nothing tof ormally define structures themselves 11:09:55 i offered a very simple formal definition: structures are just ordinals 11:10:09 but okay. so you dont want to accept that structures are odinrals. 11:10:12 what are they then? 11:10:23 how do you expect us to argue about strcutres if we cant even fucking agree on what they are? 11:11:20 and don't give me this "it's intuitively obvious" shit. we've already disagreed on what structures are meant to be, so it should be obvious as hell that they are NOT infact induitiviley obvious 11:11:22 they are ordered sets of entries 11:11:31 that's isomorphic to ordinals so ... close enough 11:11:31 what's an entry 11:11:39 That's hardly the point of disagreement 11:11:46 in fact it is 11:11:53 LORD 11:12:01 what's an entry?! It's an argument 11:12:06 of a function 11:12:12 LEt me ask another question: 11:12:13 It's the simplest thing to define 11:12:20 Your entries are ordered, by what? 11:12:30 What are indices of possible positions? 11:12:34 that's why I use "argument" instead of "entry". Because it's the standard term in professional mathematics 11:12:48 If you prefer you can also use "parameter" from computer science 11:13:04 are you going to tell me these are vague ill-defined terms? 11:13:10 i define an entry as an ordered pair consisting of an ordinal and the value that it maps to. 11:13:25 i'm asking these "dumb" questions because we apparently don't agree that arrays are based on ordinals 11:14:28 do you believe in order-types? 11:14:43 What order types do you mean now? 11:14:46 why are you asking the obvious things. You can order them using standard ordinal notations if you like 11:14:54 i define an array as a function mapping from e0 to the positive integers, such that only a finite number of entries are greater than one 11:14:57 the usual meaning! 11:15:11 okay so we DO agree on the definition of arrays that i provided? 11:15:15 ordinals and other things which can be ordered 11:15:46 I agree that the approach of indexing entries by ordinals is handy 11:15:52 so ... nothing beyond e0? 11:15:57 but not sufficient to define legion space in and of itself 11:15:57 cookiefonster: just for now 11:16:12 you can't even define tetrational arrays with that alone 11:16:15 ok, so that's up to tetrational arrays 11:16:20 SbiisExE: so you don't agree that it works in the long term 11:16:25 then what would work instead? 11:16:26 but you can create array notations with it 11:16:29 no 11:16:39 It works as a basic framework 11:16:47 look BEAF works on two levels 11:16:56 Brb 11:17:05 on one level, we have the 5 or so fundamental laws of BEAF 11:17:17 That define what to do in the basic situations 11:17:27 ie. if there is less than 3 entries 11:17:33 if prime=1 11:17:45 If the pilot is at a successor entry or limit entry 11:17:46 etc. 11:18:01 Then you start defining your spaces using ordinal indexed entries 11:18:24 The only thing left to figure out then is, what happens when the pilot is at limit ordinals 11:18:46 So I agree that underlying Bowers' theory is ordinals, whether or not he realized it 11:18:59 brb 11:19:08 the problem is, if we talk about stuff like tetrational arrays of legion arrays 11:19:26 then we have to think about how those "spaces" translate into sizes under Bowers' scheme 11:19:40 and there's the problem: bowers' scheme is vaguely defined 11:19:41 this requires another level which is not simply ordinal related 11:19:52 It's fundamental sequence related 11:20:00 so you alter the fundamental sequences. there. 11:20:01 It's about the choices that are made 11:20:07 yes 11:20:12 what's the problem then 11:20:25 but clearly you shouldn't alter anything which Bowers' already established 11:20:38 except bowers' didn't establish shit 11:20:51 and also, one can see implicit rules based on what Bowers' has already provided 11:20:57 @Vel fundamentally disagree with your definition. It uses an approach which is not inline with the climbing method, and has, once again, more to do with ordinal arithmetic 11:21:21 can I get everyone to agree that an array of structure A, has p times the number of entries for structure A? 11:21:41 (most likely no one is going to even understand what I mean except cf) 11:21:59 (and it's a perfectly simple and well-defined concept btw) 11:22:04 i'm trying to figure out what this is supposed to mean? 11:22:08 what's an array of structure A? 11:22:11 right 11:22:15 so let me break it down 11:22:34 let's just say, for the sake of the argument, that some subsystem of BEAF is understood 11:22:42 i get what you mean 11:22:48 so let {b,p (A) 2} diagonalize over that system 11:23:10 and say it creates { A(p)&b } 11:23:28 where A(p) is a function of p. A(p) is the number of entries that result 11:23:58 somehow it also codifies the entries in the structure that get non-1 values, and which don't (btw this can also be defined in sound ways at least up to e0) 11:24:18 Now create a new delimiter, called (A+1) 11:24:24 can we agree that 11:25:00 {b,p(A+1)2} = { A(p)&b (A) A(p)&b (A) ... (A) A(p)&b } w/p copies of A(p)&b 11:25:10 and that this would have exactly p*A(p) 11:25:16 entries 11:25:39 essentially you are taking a structure A and asking about A*w 11:25:47 right 11:26:02 so the 2 would now be in position A*w 11:26:08 Is that not clear? 11:26:11 with any extension of the Wainer hierarchy, yes that is true 11:26:17 w/e 11:26:23 were talking about BEAF 11:26:33 but I understand that means you agree 11:26:50 because the ordinal definition of BEAF so far uses the wainer hierarchy 11:27:10 that's part of the parameters spelled out by Bowers' writings 11:27:11 will be afk, gotta rake leaves 11:27:23 so simply defining everything = 1 is not in line with this 11:27:44 But why can't this extend to ANY ordinal? 11:28:22 clarify 11:28:34 If we agree to subsystem up to ordinal A 11:28:45 then we already have a definition up to ordinal A*w 11:28:46 no? 11:29:12 (well technically you need A(p)&b first which is something distinct from the subsystem up to A) 11:29:23 (assume we have already chosen such a diagonalization) 11:29:47 sure 11:29:52 okay 11:29:59 so there is at least one thing we agree one 11:30:01 on 11:30:24 Of coarse this doesn't really get us very far 11:30:35 we can only iterate this up to A*w^w 11:30:55 but that should also be pretty clear since Bowers' shows us how dimensional arrays work 11:31:13 which means we can have dimensional arrays of A-blocks, as opposed to just entries 11:31:51 sure 11:31:56 You say Bowers' didn't establish shit 11:32:00 but that's not true 11:32:11 that's why we can more or less agree up to e0 11:32:32 and there is enough to go on to get to e0*w^w based on what we agreed on already 11:32:38 @Vel fundamentally disagree with your definition. It uses an approach which is not inline with the climbing method, and has, once again, more to do with ordinal arithmetic 11:32:43 that's why we can more or less agree up to e0 11:32:47 So it follows that if A is known, then subsystem A*A is also known 11:33:05 I meant past e0 11:33:20 up to e0 it should be fine if your just using the wainer hierarchy 11:33:50 Next we can nest structure A to get to A^X 11:34:02 is there something equivalent to A^X in your notation? 11:34:31 like, can I take any epsilon number and have {{w,w,2},w} ? 11:35:49 my definition does not define BEAF over ordinals 11:35:55 ? 11:36:01 it does not allow ordinal entries 11:36:15 if by that you mean e0^w, then no, my definition does not cover that 11:36:33 well there's the problem then 11:36:48 um. it's delibereately designed to only go op to e0 11:36:48 undoubtably you use w^(e0*w) instead 11:37:16 I distinctly remember you originally defined it to go to legion space 11:37:27 which you believed was LVO according to your notation 11:37:35 have you altered it since then? 11:37:37 that was a conjecture according to the work of iko/wyth/etc. 11:37:47 later i just gave up on it after finding a critical error 11:38:09 i decided that it's just not worth my time. bowers can formalize his own god damn notation 11:38:23 sure 11:38:34 or other interested parties can pick up where he left off 11:39:06 It's not as if your "doing his work for him" 11:39:32 If someone defines something useful out of BEAF then they deserve credit for helping to formalize the idea 11:39:47 i realized that there was really no point in what i was doing 11:39:52 should we just disregard all of Chris Bird's work because ... he's just working with Bowers' arrays 11:39:55 of coarse not 11:40:05 he is rightly recognized for his contribution 11:40:15 and Bowers' was definitely helped out by his work 11:40:27 if bowers didn't bother to create a formal notation, then the emperor wears no clothes and i'm not patching them up for him 11:40:32 none the less Bowers' still get's credit for the initial idea 11:41:12 What he has is a sloppy idea 11:41:15 not no idea at all 11:41:17 that's all 11:41:28 But you basically want to say he's got nothing 11:42:20 that he should be disregarded and ignored 11:42:26 but I don't understand that 11:42:36 I much prefer Bowers' as a founding father over Joyce 11:42:39 what i mean to say is that the only reason i was trying to formalize his notation 11:42:45 at least his vision has far more soaring 11:42:56 was that BEAF has been put on a pedestal 11:42:59 and ... actually a lot better defined, despite it's considerable limitations 11:43:12 yes 11:43:15 and from trying to fix it, i realized...why is it even being put on a pedestal at all 11:43:18 this is undeniably true 11:43:27 should we dethrone him? 11:43:29 why should i care about this half-broken notation 11:43:41 by simply saying "it's all ill-defined" and amounts to nothing 11:43:44 SbiisExE: i said nothing about discounting ALL of bowers work, including his illions 11:43:49 doesn't seem very googological in spirit 11:44:09 if his notation offered something new or genuinely interesting, then i would be more inclined to try to fix it 11:44:20 let's just disregard any work with potential. Then we don't ever have to try and outdo it, in either percision or scale 11:44:39 except i would argue that BEAF has no potential 11:44:56 it does thogh 11:45:15 it does though. He independently discovered a lot of things and put an interesting twist on it 11:45:26 such...as...? 11:45:39 @Vel BEAF has no potential. That's nonsense 11:45:43 of coarse it has potential 11:45:54 Back 11:45:58 It's got as much potential as trying to get SGH to catch up to FGH 11:46:00 at least 11:46:27 SGH, HH, and FGH are extremely simple and well-defined ways to generate large numbers on virtually unlimited scales 11:46:30 I don't get the comparison 11:47:03 and so is BEAF 11:47:04 BEAF is a clunky, half-broken system that is at most as powerful as the above three 11:47:14 it's no more or less ill-defined than SGH, FGH, or HH 11:47:24 fuck no 11:47:29 we can't even agree on the definition of a structure 11:47:31 fuck yes 11:47:37 What's FGH? 3 rules 11:47:43 but we can agree on the definition of an ordinal 11:47:45 would you say it's well defined? 11:47:50 I'd say so 11:48:00 so why is there a problem applying that to BEAF? 11:48:11 we can't even agree on the definition of a structure 11:48:19 FGH, SGH, HH just need fundamental sequences to run 11:48:23 because we can't agree on the basic definitions of the fundamentals of BEAF because bowers didnt even bother to define those 11:48:26 and BEAF just needs structures defined 11:48:32 It's perfectly well defined 11:48:35 Then define the structures for god's sake 11:48:37 and by "funcamentals" i mean not FS's but things like entries and shit 11:48:41 The only problem is in the 2nd layer 11:49:10 Chris Bird did just fine taking array notation very far 11:49:24 i haven't really bothered to look over his work because it's massive 11:49:29 Same 11:49:32 of coarse he NEVER talks about array-sizes, or entry counts. That's the 2nd layer 11:49:46 Bird took a way around 11:49:49 and i'm skeptical considering that he started the entire fad of going "is comparable to f_a(n)" which is a massive red flag for me 11:49:55 But the first layer of BEAF is just as well defined as FGH,SGH, or HH 11:49:56 So he didn;t have to fiddle with all the Bowers' crap 11:50:37 you say we can't agree on the definition of a structure 11:50:41 but we already did 11:50:46 An ordered set of entries 11:50:51 First layer, possibly 11:50:51 that's no problem 11:50:54 @Vel fundamentally disagree with your definition. It uses an approach which is not inline with the climbing method, and has, once again, more to do with ordinal arithmeti 11:50:59 But the problem with BEAF is the second layer 11:51:04 Which FGH doesn't have 11:51:04 so? 11:51:13 how does argue with what I said 11:51:25 we do not agree on my definition up to e0 11:51:42 How does climbing method magically now agree with vell's definition? 11:51:48 Let me be more explicit. Your continuation past e0, which you originally posted, does not follow Bowers' guidelines that structures should only be named according to the growth rate of entries 11:51:51 for that particular space 11:52:00 very specific complaint 11:52:10 Growth rate of entry? 11:52:12 whats the growth rate of an entry, help 11:52:15 It's not a blanket complaint to your entire approach to BEAF, so stop painting it as such 11:52:30 sigh 11:52:32 I'm sorry 11:52:39 I can't talk to you guys about this 11:52:47 There just isn't the terminology needed at this point 11:52:48 Because...? 11:53:02 and that's a result of the fact that bowers' writings are vague 11:53:05 Bowers' didn't create one 11:53:15 I mean he has terminology but it's used loosely 11:53:28 I don't think it's that hard to make sense of 11:53:35 we have these differing interpretations and we don't agree on things because bowers did a crap job in defining notation 11:53:51 this i consider evidence for the view that BEAF does not deserve the respect it has 11:53:53 but if you guys need me to explain every term "entry" "structure" "array" "space" "size" etc. 11:54:10 then this is just going to be a boring explication of elementary terms 11:54:27 Well 11:54:27 We agree 11:54:33 we just don't realize it 11:54:34 If you want BEAF well-defined 11:54:43 You would have to define all the terms 11:55:00 If you understood what I meant you wouldn't be acting like I'm talking nonsense 11:55:14 the basic ideas are pretty simple. But the consequences are hard to work out 11:55:21 " you wouldn't be acting like I'm talking nonsense" thanks for implying that we're being dishonest 11:55:41 which is why Bowers' never really understood the beast he unleashed when he tried to imagine legion space 11:55:47 I'm just pointing out what I think is required for formal definition 11:55:57 I not saying your being dishonest 11:56:06 I think your bafflement is genuine 11:56:16 but for whatever reason the language being used here is a barrier 11:56:18 fair enough 11:56:58 I think Bowers' gave enough of a framework to generate a formal definition 11:57:07 at least for variants of what we might call BEAF 11:57:31 the only difficultly is in gauging the "sizes" (theres that technical term again) of the structures generated 11:57:53 then this is just going to be a boring explication of elementary terms 11:58:00 right 11:58:06 without these elementary terms, we get into discussions like the above 11:58:12 that's WHY we define them 11:58:24 I'd rather avoid having to explain these terms. I thought they'd be well understood to anyone who has spent time trying to understand BEAF 11:58:33 You shouldn't have to ask me what an entry is, at least 11:58:49 And a structure, while vaguely discussed in Bowers' own writing 11:59:01 clearly refers to an ordered set of entries 11:59:31 as bowers says there are linear structures, planar structures, dimensional structures, structures that need tetration to define , etc. 11:59:59 What all these and more have in common is that the entries can always be put in order, in one-to-one correspondence with some subset of the countable ordinals 12:00:16 So what's so unclear about the term "structure" then 12:00:18 BONG BONG BONG BONG BONG BONG BONG BONG BONG BONG BONG BONG 12:00:35 who cares what he calls it, tetrational-structure, X^^X-structure 12:00:52 The underlying idea is still an ordered-set of entries of order-type e0 12:00:58 Is that ill-defined? 12:01:46 did bowers ever explicitly specify this this? 12:02:03 That is one genuine problem 12:02:13 that is the genuine problem with the entirety of beaf 12:02:20 Bowers' entire work is written in an informal style 12:03:01 But (1) that doesn't mean he is talking nonsense. Just that he is not the best communicator (2) It doesn't mean his idea is inherently wrong 12:03:22 But I really don't understand why you object to such a simple definition 12:03:36 to me it seems very clearly in the spirit of what Bowers' was writing about 12:03:41 its communicated in a vague enough way that there is "room for interpretation" 12:03:47 which is okay in art, and bad in mathematics 12:03:50 He obviously thinks of structures as "spaces" made up of entries 12:04:29 and the easiest way to make sense of them is to order the entries using ordinals rather than his more esoteric idea of literally listing the entries out in multiple dimensional space and beyond 12:05:21 He also clearly thinks that there is a structure/space corresponding to every dimension we could define using the powers of w 12:05:44 But I don't think you actually disagree with me 12:05:53 We're just saying the same thing in different ways 12:06:40 we both intuitively understand how BEAF is "supposed" to work 12:06:58 and i'm sure we both agree that the three main rules of beaf are clearly specified enough 12:07:20 ...given the foundations of how arrays are defined 12:07:22 okay 12:07:40 I dislike his definition of dimensional arrays btw 12:07:53 It's ridiculous that he needs that many words to define it 12:08:10 but that is because he didn't even bother to have a writable notation past planar arrays 12:08:16 and your stance on "arrays are functions mapping ordinals to positive integers" is not 100% ckear to me but i think it's agreement? 12:08:18 because the entries would be "in another dimension" 12:08:52 Chris Bird's approach is much more practical. In line everything, treat the "space" as a manner of speaking, and use delimiters to denote the same concept 12:08:56 (I think we need it so that all but finitely many ordinals get mapped to 1) 12:09:10 Wojowu: i left that out to save typing 12:09:31 well I wouldn't agree with that 12:10:01 It's a weird function which takes number-ordinal pairs and maps it to a number 12:10:17 i think you misunderstood me 12:10:20 and of coarse it's not like a regular function, because it can take as many or as few arguments as you like 12:10:21 Why should it take pairs? 12:10:32 The position and value of each entry 12:10:38 is an ordinal-number pair 12:10:43 an array itself -- separate from the "v(A)" function -- is itself a function mapping ordinals to positive integers 12:10:48 w/ the restrictio wojwo mentioned 12:10:51 take a few of these pairs and they define an array to solve 12:11:03 No, an array would be a function from ordinals to numbers 12:11:03 I guess 12:11:05 beaf is a function that maps arrays to positive integers 12:11:13 but it's not the way I usually think about it 12:11:20 So that, on input (position) we get output (value at the position) 12:11:32 you mean like {3,3,3} --> w^2*3+w*3+3 , right? 12:11:36 no 12:11:59 {3,3,3} is the following function: f(0) = 3, f(1) = 3, f(2) = 3, f(3) = 1, f(4) = 1, .... f(w) = 1, ... 12:12:14 what? 12:12:23 oh 12:12:38 I don't see why your defining such a function as BEAF 12:12:50 this is the array itself, separate from evaluation 12:12:54 sure, that may be part of how BEAF works 12:13:04 but I'm talking about {3,3,3} mapping to a value 12:13:06 tritri is what you get when you plug this "f" into the evalution function "v" 12:13:08 of 3^^^3 12:13:19 k 12:13:24 fine 12:13:26 vell described a formal way to talk about arrays 12:13:31 k 12:13:39 do you have a better definition you suggest? you seem skeptical 12:13:49 like I said, we agree on the fundamentals. 12:14:00 You guys are mainly objecting to the terminology 12:14:35 But there is nothing wrong in what I said either 12:14:49 BEAF takes a set of ordinal-number pairs and returns a number 12:14:57 It's just a way of looking at it 12:15:24 "set of ordinal-number pairs" is isomorphic to "function from ordinals to numbers" 12:15:32 so there is no problem 12:15:35 and your stance on "arrays are functions mapping ordinals to positive integers" is not 100% ckear to me but i think it's agreement? 12:15:38 so the answer to this is yes 12:15:41 how can I be "wrong" then and you "right" 12:15:41 gotcha 12:16:05 Except that what you said allows things like (w,1) and (w,2) at the same time, but it;s technicality 12:16:15 right 12:16:20 I realized that of coarse 12:16:29 so from my definition, we can easily define what entries and structures are 12:16:46 okay 12:16:53 entry in array A is a pair (a,n) such that A(a) = n 12:16:55 except that you can't as I said earlier 12:17:15 ? 12:17:17 structure is just an ordinal in the domain of A 12:17:21 This only formalizes the first layer of Bowers' idea 12:17:36 as long as we don't care how big the arrays get 12:17:39 then it's no problem 12:17:47 but try to figure out when we hit pentational arrays 12:17:55 and this idea, in and of itself, is insufficient 12:18:03 This that clear to you? 12:18:22 it is not, in fact 12:18:29 Well, the idea of just having this v(A) function is not enough for anything 12:18:35 Unless we define how v works 12:18:36 what stops us from defining pentational arrays w/ ordinals? 12:18:38 this idea cant not define pentational arrays 12:18:53 Why so? 12:19:03 We can have them as ordinals below zeta_0 12:19:09 because of the disconnect between the number of entries and the ordinal expression you use 12:19:35 This all collapses to defining fundamental sequences, I'd believe 12:19:45 Maybe so 12:19:53 the wikipedia article on the veblen hierarchy has defn's for FS's up to Gamma_0 12:19:54 but no one has successfully done so 12:20:09 all the sequences so far arbitarily SAY what the size of an array is 12:20:18 instead of caring about what it actually is 12:20:21 what is the size of an array 12:20:32 I will try one more time 12:20:35 Number of non-1 entries, I'd say 12:20:42 please, if you understand it, stop asking me the question 12:20:46 thanks 12:20:54 Let A be an ordinal 12:20:59 assume it's a limit ordinal 12:21:19 {b,p(A)2} produces A(p) entries = to b 12:21:33 What is A(p)? 12:21:36 that function A(p) is the size of the structure 12:21:45 Okay 12:21:50 a function which returns the number of entries 12:22:07 You lost me 12:22:09 for example {b,p(0,1)2} would be the function A(p) = p^p 12:22:13 so you mean |Pi(a)| as defined here http://googology.wikia.com/wiki/Introduction_to_BEAF#As_ordinals_.28advanced.29 12:22:19 right 12:22:21 basically 12:22:25 see what I mean 12:22:25 Ah, okay, I see 12:22:33 you already know what I mean, you just don't know it 12:22:53 whatever yogi berra :P 12:22:55 But, 0,1 isn't an ordinal 12:23:06 meaningless point 12:23:15 w 12:23:17 ^w 12:23:24 That's better 12:23:25 Any set which can be ordered can be treated as an ordinal notation 12:23:33 False 12:23:42 how is that false 12:23:49 sigh 12:23:50 Set of real nmbers can be ordered :P 12:23:54 you guys love the technicalities 12:24:00 Of course we do 12:24:06 thats why we do math 12:24:18 well anyway, you know perfectly well that has nothing to do with 0,1 as an ordinal notation 12:24:32 Indeed 12:24:38 okay 12:24:44 it should be reasonably easy to map bowers delimiter notation to ordinals, but im too lazy to bother 12:24:53 So how do we define ordinals as opposed to order-types then? 12:24:58 this is a genuine question 12:25:17 um, an ordinal is an order type of a well-ordered set? idgi 12:25:24 You want a formal defintiion? 12:25:28 vell gave you one 12:25:47 well, I need a better understanding of well-ordered to make sense of that 12:25:58 !w well-order 12:26:03 !wp well-order 12:26:04 http://en.wikipedia.org/wiki/well-order 12:26:17 In well-ordered set, it's impossible to have infinite descending chain 12:26:29 okay 12:26:31 good enough 12:26:36 So, for example, integers are not well-ordered, because we have 0>-1>-2>... 12:26:51 But natural numbers are well-ordered 12:27:08 So then I can say, any well-ordered set, can represent an ordinal notation 12:27:13 " In well-ordered set, it's impossible to have infinite descending chain" this is dependent on the axiom of dependent choice 12:27:20 but we're in zfc i'm assuming :P 12:27:38 Well, maybe 12:27:54 I could understand the hesitation 12:28:15 But anyway, the point is, that I often refer to any well-ordered set as an ordinal notation 12:28:26 Uhh 12:28:27 So to me it's clear that (0,1) is an ordinal notation 12:28:35 um 12:28:38 What is ordinal notation for you? 12:28:40 what now 12:28:48 A way to label ordinals? 12:29:02 why, your going to know give me the technical definition of an ordinal notation? 12:29:07 "a partial function from finite strings in a finite alphabet to ordinals" 12:29:08 In that case, well-ordered set doesn't give you ordinal notation 12:29:34 Because if I say "set of countable ordinals" I give you no way of labelling ordinals 12:29:42 A well-ordered set can always be put into one-to-one correspondence with some subset of the ordinals, no? 12:29:51 god 12:30:09 You guys ever considered that some things should be implied 12:30:15 and that nitpicking wastes time 12:30:29 But I honestly don't see what you mean 12:30:31 Obviously I was only thinking of stuff that can be bounded by a countable ordinal 12:30:34 i honestly dont get what you mean by "I often refer to any well-ordered set as an ordinal notation" 12:30:37 Okay them 12:30:48 "set of ordinals below epsilon_0" 12:30:53 I don't see what's confusing 12:30:59 Does this set, per se, give you a way of labelling ordinals? 12:31:00 ExE uses a set of delimiters 12:31:05 that can be well-ordered 12:31:17 and so #^^^# is just another ordinal notation for gamma(0) 12:32:11 Okay, maybe we can put this aside for now 12:32:15 Back to BEAF 12:32:20 From what I can see so far 12:32:38 Only thing we have yet to say is how function A(p) looks depending on A 12:33:02 sure 12:33:12 Wait, there is something else 12:33:13 and on how we might chose to select entries 12:33:24 A(p) only tells us how many entries there will be 12:33:32 Not where they will be in new array 12:33:37 ie. on an ordinal and something akin to it's fundamental sequence 12:33:46 sure 12:33:56 but it's easy to set it up in the definition to define this as well 12:34:10 Yes 12:34:17 For example X&(b,p) = b,b,b,...,b w/p bs 12:34:38 Now now, we only have to do so for all ordinals we care about 12:34:59 X^2&(b,p) = X&(b,p) (1) X&(b,p) (1) X&(b,p) (1) ... (1) X&(b,p) w/p X&(b,p)s 12:35:20 ie. treat it like a string 12:35:33 this makes the operations very simple and mechanical 12:35:34 Can you please use ordinals instead of X's? 12:35:54 ordinal positions and multi-dimensional spaces can then be thought of an theoretical interpretations 12:36:00 why? 12:36:04 we are talking about BEAF 12:36:14 Well, I thought we are trying to define BEAF in terms of ordinals now 12:36:17 so I assume we are trying to define the Xs 12:36:29 Or do that 12:36:30 Ordinal notation are irrelevant? 12:36:39 They actually are 12:36:42 ordinal notations are irrelevant? 12:36:45 I believe 12:36:49 why do we have to keep returning to the same points 12:36:56 ? 12:37:00 why do they matter? 12:37:16 Because they give us a way to write ordinals down 12:37:22 (I just got through discussing how countable well-ordered sets can be used as ordinal notations) 12:37:27 For example, w^w is an ordinal notation for some ordinal 12:37:27 of coarse 12:37:36 but using #^# or w^w or X^X shouldn't matter 12:37:43 so why bring it up? 12:37:57 Because there isn't always an obvious isomorphism 12:38:07 What should be zeta_0 in X's? 12:38:26 If we use ordinals, we should stick to ordinals 12:38:32 also using something more esoteric like sigma(0,sigma(0,1)) 12:38:35 And their standard notations 12:38:43 w/e 12:38:57 if ordinals are something that transcend notation then what you are saying is pedantic 12:39:06 we are sticking to ordinals regardless of what notation is used 12:39:22 what you mean is you want to stick to the standard notation 12:39:29 I don't see why that matters so much 12:39:43 Actually, ordinals transcend all notations, but it's not the point 12:40:02 knowing the definition of gamma(0), SVO, LVO, BHO, etc. isn't going to help us define BEAF structures. It's not going to tell us what X^^^X is for example 12:40:08 k 12:40:11 True 12:40:14 what's the point then 12:40:20 But can you prove that X structures are well-ordered? 12:40:30 How do you order them, first of all? 12:40:33 can you prove w is well-ordered? 12:40:43 0,1,2,3,4,5,... etc. 12:41:00 If we define w in set-theoretic terms, yes I can Category:Blog posts